Formulas and reference

Sum of angles in a triangle

\[\alpha+\beta+\gamma=180^\circ\]

Semiperimeter of a triangle

\[p=\frac{a+b+c}{2}\]

Triangle area from a side and altitude

\[S=\frac12ah_a\]

Triangle area from two sides and the included angle

\[S=\frac12bc\sin\alpha\]

Heron's formula

\[S=\sqrt{p(p-a)(p-b)(p-c)}\]

Law of cosines

\[a^2=b^2+c^2-2bc\cos\alpha\]

Law of sines

\[\frac a{\sin\alpha}=\frac b{\sin\beta}=\frac c{\sin\gamma}=2R\]

Pythagorean theorem

\[a^2+b^2=c^2\]

Cosine of an acute angle

\[\cos\alpha=\frac bc\]

Sine of an acute angle

\[\sin\alpha=\frac ac\]

Tangent of an acute angle

\[\tg\alpha=\frac ab\]

Area of a parallelogram

\[S=ah_a\]

Parallelogram area from two sides and an angle

\[S=ab\sin\gamma\]

Area of a rectangle

\[S=ab\]

Rhombus area from diagonals

\[S=\frac12d_1d_2\]

Area of a trapezoid

\[S=\frac{a+b}{2}\,h\]

Circumference

\[L=2\pi R\]

Area of a circle

\[S=\pi R^2\]

Equation of a circle

\[(x-x_0)^2+(y-y_0)^2=R^2\]

Midpoint coordinates in space

\[x_0=\frac{x_1+x_2}{2}\]

\[y_0=\frac{y_1+y_2}{2}\]

\[z_0=\frac{z_1+z_2}{2}\]

Distance between two points in space

\[AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\]

Coordinates of vector AB

\[\overrightarrow{AB}=(x_2-x_1;\,y_2-y_1;\,z_2-z_1)\]

Dot product in coordinates

\[\vec a\cdot\vec b=a_1b_1+a_2b_2+a_3b_3\]

Dot product using the angle

\[\vec a\cdot\vec b=|\vec a||\vec b|\cos\varphi\]

Volume of a right prism

\[V=S_{\rm осн}H\]

Lateral surface area of a right prism

\[S_{\rm біч}=P_{\rm осн}H\]

Volume of a pyramid

\[V=\frac13S_{\rm осн}H\]

Lateral surface area of a regular pyramid

\[S_{\rm біч}=\frac12P_{\rm осн}m\]

m — apothem.

Volume of a cylinder

\[V=\pi R^2H\]

Lateral surface area of a cylinder

\[S_{\rm біч}=2\pi RH\]

Volume of a cone

\[V=\frac13\pi R^2H\]

Lateral surface area of a cone

\[S_{\rm біч}=\pi RL\]

Volume of a sphere

\[V=\frac43\pi R^3\]

Surface area of a sphere

\[S=4\pi R^2\]